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Computing Interleaving and Bottleneck Distance for 2-D Interval Decomposable Modules

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Computing Interleaving and Bottleneck Distance for 2-D Interval Decomposable Modules
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19
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CC Attribution - NonCommercial - NoDerivatives 4.0 International:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
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Abstract
Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For 1-D persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most n-D persistence modules, n>1, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called {\em 2-D interval decomposable} modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the interleaving distance between two such indecomposables. This leads to a polynomial time algorithm for computing the bottleneck distance between two 2-D interval decomposable modules, which bounds their interleaving distance from above. We give another algorithm to compute a new distance called {\em dimension distance} that bounds it from below.